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DOI : 10.2240/azojomo0268

Co Diffusion Along Alpha-Zr and Beta-Zr 20% Nb Grain Boundaries

C. Corvalán Moya, M. J. Iribarren and F. Dyment

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This is an AZo Open Access Rewards System (AZo-OARS) article distributed under the terms of the AZo–OARS https://www.azom.com/oars.asp which permits unrestricted use provided the original work is properly cited but is limited to non-commercial distribution and reproduction.

AZojomo (ISSN 1833-122X) Volume 4 July 2008

Topics Covered

Experimental Procedure
Results and Discussion
   Kinetic Type B
   Kinetic Type C
Contact Details


Atomic transport along grain boundaries (GB) and interphase boundaries (IB) plays a key role in a large number of metallurgical processes.

The experimental determination of GB or IB diffusion coefficients is very similar to that for the corresponding bulk diffusion, the main difference being the relation between penetration and diffuser concentration. In this framework, Harrison’s classification allows us to deal with different kinetics (type A, B and C). The proper employment of type B and C kinetics in different ranges of temperature provides the opportunity to obtain not only extended diffusion parameters, but additional information about the physical factors included in the “apparent diffusion coefficient” in grain boundaries (P = s Dgb).

In this paper, diffusion parameters of cobalt along the grain boundaries of pure α-Zr are presented. In order to cover a range of temperature of technological interest, the interval 430-633 K was studied. Suzuoka and Gaussian solutions for type B and C kinetics were employed. Also, diffusion parameters of cobalt along grain boundaries of ß-Zr 20% Nb are presented in a temperature interval of 710-875 K, according to B kinetics.

The results show that cobalt behaves as an ultra-fast diffuser in α-Zr and ß-Zr 20% Nb grain boundaries. In α-Zr, comparison between B and C kinetics allows us to estimate the segregation factor, its temperature dependence and hence, to obtain the true value of Dgb even in the range of temperature where only P is available.


Diffusion, Pure Zr, Zr-Nb Alloys, Grain Boundaries, Segregation


Atomic transport along both grain boundaries (GB) and interphase boundaries (IB) is orders of magnitude faster than in the bulk crystal. For this reason both GB and BI play a key role in a large number of metallurgical processes such as plastic deformation, corrosion at high temperatures, solid-state transformations, surface treatments and the stability of precipitates in a matrix.

Zr-based alloys (mainly Zircaloys and Zr 2.5% Nb) are extensively used in nuclear and chemical industries due to their excellent corrosion resistance, mechanical properties and nuclear properties. They are usually employed in polycrystalline forms, exhibiting a high density of GB and IB.

In this context, solute and self diffusion studies in the bulk and in GB and IB are relevant from a basic and an applied point of view. In most nuclear power installations, Fe, Co, Ni and Cr are normally present as solutes or impurities in devices which are in contact with Zr. For instance, different types of steels, including stainless steels, are used along with Zircaloys or Zr-Nb alloys. Elements such as Fe, Co, Ni and Cr show diffusion coefficients several orders of magnitude faster than self-diffusion in Zr and they are usually referred as ultra-fast diffusors.

The experimental determination of the diffusion coefficients in GB and IB is similar to that for bulk diffusion, except that the relation between the concentration of the diffusor and the penetration is approximately linear instead of parabolic.

Experimental Procedure

The material was cut into samples around 0.75 cm2 surface area. They were mechanically polished finishing with 1 micron diamond paste. Some of the samples were employed for metallographic purposes in order to correlate the real morphology with the mathematical models, for the correct application of the different possible kinetics. Zr-20%Nb samples were annealed at 1223 K for 2 hours, and then quenched, obtaining a stabilized grain boundary network structure, while pure α-Zr samples showed a well stabilized structure of small grain size.

The radiotracer 60Co was obtained in the laboratories of the National Atomic Energy Comission of Argentina.

Cobalt powder was irradiated in the Nuclear Reactor RA1; the irradiation time was calculated in order to obtain the minimun activity suitable for our experiments. For the dissolution of the Co two different reagents were assessed: nitric and hydrochloric acids. As a first attempt, non irradiated Co was used. According to this experience, nitric acid was chosen since it generates a minor amount of active solution and the dissolution is completed in a shorter time. In addition, according to data obtained from the bibliography, the use of nitric acid in preference to hydrochloric acid diminishes the damage caused by corrosion of the samples, which, if it happens, would disturb the diffusion results. The 60Co was obtained, therefore, in a HNO3 solution and it was deposited directly onto the polished surface of the samples in order to make a diffusion couple. The samples were wrapped in high purity tantalum foil and vacuum sealed in quartz tubes with a slight overpressure of high purity argon. The diffusion anneals were performed in Adamel electrical furnaces, provided with P+I+D electronic control, with a precision of ± 1 K.

Later, the samples were turned radially, to assure a flat front diffusion from the deposited face. The sectioning of the samples was performed in a precision abrasion machine. The thicknesses of the removed layers were determined by weight difference of the samples after each sectioning. The measurements were made with a 10-5g accuracy electronic balance; the density of the material involved and the area of the samples were known.

An INa(Tl) well detector was employed in order to measure the activity (concentration) of the layers, this information being collected and analyzed by means of a multichannel analyzer device.

The quoted errors were estimated around 15% for the grain boundary diffusion parameters Pgb and Dgb.

Results and Discussion

Diffusion in grain boundaries is a complex process that includes the serial elemental processes:

  • bulk diffusion from the surface of the sample
  • diffusion along the GB
  • bulk diffusion from the GB

According to the relative importance of these processes, different situations or kinetics could result. Harrison [1] allows us to deal with three different kinetics: types A, B and C. We will give the expressions corresponding to B and C kinetics, both used in this work.

Kinetic Type B

Main processes: diffusion along GB and bulk diffusion from GB.
100 d < bulk diffusion distance = (Dt)1/2 < d/20
where d is the mean grain size and a is the GB thickness.Then we can apply Fisher’s model [2] of an isolated GB.
In this framework, and in order to obtain the penetration profile, the following equations are applied:

Constant source


Instantaneous source


where s is the segregation factor; D is the bulk diffusion coefficient and Pgb = δ Dgb is the experimentally measurable parameter, called “apparent diffusion coefficient“. The B kinetic is the more employed one because it allows reasonable times and temperatures for the diffusion anneals.

Figure 1. Schematic illustration of diffusion B kinetic.

Kinetic Type C

Main process: diffusion along GB. The bulk diffusion from the surface and from the GB is meaningless.

The penetration profile in this kinetic, equivalent to bulk diffusion, is either gaussian (instantaneous source) or an error function (constant source). Dgb is directly measured. For a Gaussian solution

C(x, t) = Co exp (-x2/4Dgbt)       (3)

where Co is the initial diffusor concentration.

Figure 2. Schematic illustration of diffusion C kinetic.

Figure 3 shows a typical penetration profile in B kinetic obtained in α-Zr GB.

Figure 3. GB Diffusion in α-Zr. B Kinetic.

The temperatures and annealing times of the experiments done with this kinetic were determined using data from Vieregge and Herzig [3]. The range of temperature used in this work was very low and the maximum mean square bulk diffusion distance calculated was 0.4 µm. As the grain size was between 7 and 10 µm, it was possible to consider that we have isolated grain boundaries, necessary for the B kinetic. Experiments at the same temperature for different times were considered unnecessary as the concentration profiles followed well the model used. It is important to remark that the structure of the α-Zr samples remains unchanged during the diffusion annealing. This is so because a possible change, for instance grain size, would be related to Zr bulk self-diffusion. According to previous work, Co [4] is a diffusor about nine orders of magnitude faster than Zr self-diffusion [5] in the bulk. As can be seen in [3], this is also similar for GB diffusion: Co diffusion along α-Zr GB is eight orders of magnitude faster than Zr GB self-diffusion in the range of temperature involved in that work. When estimating the maximum segregation of Co in α-Zr, the authors of [3] considered that about four decades of that difference have to be ascribed to different GB diffusion coefficients.
In the case of kinetic type B, Pgb are the values experimentally measured; these values are given in Table 1.

Table 1. "Apparent diffusion coefficients" in α-Zr-Zr GB



Pgb/ m3 s-1







Figure 4 shows a typical penetration profile in C kinetic obtained in α-Zr GB.

Figure 4. GB Diffusion in α-Zr. C Kinetic.

In the case of kinetic type C, the values experimentally measured are Dgb directly. The importance of this kinetics is that we do not need to know the values of the thickness of the GB and the segregation factor. The measured values in this work are shown in Table 2.

Table 2. Diffusion coeficients in α-Zr GB



Dgb/ m2s-1













In Figure 5, the Arrhenius plot shows previous experimental data [3] and results from this work: measurements in B kinetic, Pgb, and the product a. Dgb in C kinetic. It is possible to see that the influence of the segregation factor is more important at lower temperatures.

Figure 5. Arrhenius diagram of Co diffusion along α-Zr GB. B and C kinetics.

For the diffusion in Zr-20%Nb, Figure 6 shows a typical penetration profile in kinetic type B and Table 3 gives the Pgb values measured.

Figure 6.GB Diffusion in Zr-20%Nb. B Kinetic

Table 3. Apparent diffusion coefficients“ in ß-Zr 20% Nb GB



Pgb/ m3s-1
















Figure 7. Arrhenius diagram of Co diffusion along ß-Zr-20%Nb GB.


This work presents results of the diffusion of Co in GB of pure α-Zr in the range of temperatures 430-633 K and the diffusion of Co in GB of ß-Zr 20% Nb in the range of temperatures 710-875 K. In α-Zr this is the first time where the kinetic type C has been applied.

The results show that Co is an ultra-fast diffuser in GB of pure α-Zr and ß-Zr 20% Nb as it is in bulk α-Zr. In both cases the relation between GB and bulk diffusion [5, 6] is around 106, in agreement with the theory of diffusion by short-circuit paths. In both materials the Arrhenius plot is straight, indicating the operation of a unique diffusion mechanism.

In α-Zr, the measured values of Pgb in type B kinetic show a good coincidence with the values measured in [3]. The concentration of impurities, particularly Fe, is very similar in both materials (192 ppm of Fe in [3], 138 ppm in our case), the main difference being the O content, which is larger in our case.

The comparison between B and C kinetics lets us obtain a direct evaluation of the segregation factor s of Co in the α-Zr GB, according to equation (4)


For calculating s as the ratio between Pgb and d Dgb we need to take into account Vieregge and Herzig´s work at the lower temperatures; then it is necessary to consider the analysis made by the authors with respect to those values. To explain the lower activation energy obtained below 600 K, they considered the possibility of the coexistence of different diffusion paths in grain boundaries and then, the dominance of those processes with lowest activation energies. They also considered the possibility of strong segregation of an impurity such as Fe with the resultant formation of new phases in the grain boundaries that could lead to a deviation from the Henry-type segregation behaviour. In [3], the Fe content of the material used for the lower temperatures is really very low (20 ppm) as compared to the one used at higher temperatures (192 ppm). Finally they mentioned the possibility of using too large values of Co bulk diffusion, D, in the expressions used to calculate PCo. To obtain a straight line in the whole temperature range covered in [3] would imply a strong downward curvature of the Arrhenius plot of bulk diffusion of Co in α-Zr, as the square root of D appears in the expressions (1) or (2). In this work, we do not neglect the possibility of a curvature. In fact, in the work by Pérez, Nakajima and Dyment [7] this characteristic is carefully analyzed. Nevertheless, all the positive curvature of Co Pgb in α-Zr obtained in [3] would not be cancelled with the negative curvature of Co bulk diffusion in α-Zr discussed in [7]. Taking into account the former arguments and the fact that we have measured using both kinetics, a first approximation to the segregation coefficient value can be made.

The values of s at the different temperatures were calculated assuming a = 5x10-10 m. They are given in Table 4. “s” shows an important variation with the temperature, from roughly 50 to 15000 over a range of 50 K, as can be seen in Figure 8.

Table 4. Segregation factor of Co along α-Zr GB.


Pgb/ m3s-1

Dgb/ m2s-1


















Figure 8. Temperature-dependence of the GB segregation factor, s , of Co in α-Zr.

Considering the possible influence of Fe on Co diffusion and its different contents between [3] and this work, we think that a fraction of the ratio Pgb / d Dgb could not be attributed to the pure segregation factor. A small uncertainty of the s value could be assumed, taking into account that at the low temperatures Vieregge and Herzig used a purer Zr with only 20 ppm of Fe. This point deserves further investigation, and we are working in this direction.

Figure 9 shows the Arrhenius plot of Co GB diffusion in: pure α-Zr (Vieregge and Herzig´s work [3] and the present work) and in ß-Zr 20% Nb (present work); and for comparison, in α/ß IB diffusion in Zr 2.5% Nb [6]. First of all there is observed a reduction of about 4 orders of magnitude between GB diffusion in pure α-Zr and the diffusivity in GB of Zr-20%Nb and in IB of Zr-2.5%Nb. Also, it is possible to see a slight reduction of the diffusivity with the increase of Nb between IB diffusion in Zr-2.5%Nb and GB diffusion in Zr-20%Nb; this difference remains within one order of magnitude in the range of temperature measured. It is necessary to have in mind that two different kinds of short circuit paths are involved: pure ß GB in the present work and α/ß IB in [6]. A question arises: is the difference obtained related to a different diffusion mechanism along a grain boundary as compared to the other short circuit path, or is the segregation factor mainly responsible for the difference?

Figure 9. Arrhenius plot of Co GB diffusion along pure α-Zr, ß-Zr 20% Nb and a/ß diffusion in Zr 2.5% Nb.

In [6] for Co and in [8] for Cr it was interpreted that in IB diffusion in Zr 2.5% Nb mostly the a-phase acts in the complex mechanism involved in these short circuits, due to the very limited solubility of Co and Cr in the a-phase. If this is the case, it seems that for Co diffusion, the amount of Nb present in the alloy is a second order effect compared to the fact of the existence of the ß phase. This subject deserves more data to be elucidated and we are working in this direction.

Related to the difference found between Pgb in pure α-Zr compared to Pgb in Zr-20%Nb and Pib in Zr-2.5%Nb (based on Figure 9 as well as Table IV), the segregation factor could only explain between one and two orders of magnitude in the P´s difference in the region of high temperatures. So, Dgb in α-Zr could be at least two orders of magnitude higher than Dgb (in α-Zr-20%Nb) and Dib (in α-Zr/ß Zr-2.5%Nb) respectively.


This paper presents results of the diffusion of Co in GB of pure a Zr in the temperature range 430-633 K and the diffusion of Co in GB of ß Zr 20% Nb in the temperature range 710-875 K. In α-Zr these are the first results where the kinetic type C has been applied.

The results show that Co is an ultra-fast diffuser in GB of pure α-Zr and in ß-Zr 20% Nb as it is in bulk α-Zr. In both materials the Arrhenius plot is straight, indicating the operation of a unique diffusion mechanism.

In α-Zr, the results obtained for Pgb in kinetic type B show a good coincidence with the values measured in [3]. This fact and the comparison between B and C kinetics lets us to obtain a direct evaluation of the segregation factor s of Co in the α-Zr GB; extending this estimation towards higher temperatures, where only Pgb is available, the extrapolation of Dgb values (type C kinetics) seems to be applicable. Therefore, Co Dgb in α-Zr seems to be at least two orders of magnitude higher than Dgb in ß-Zr-20%Nb and Dib in α-Zr/ß- Zr-2.5%Nb respectively.


The authors wish to thank the support of CONICET (PIP n° 5322) and Agencia Nacional de Promoción Científica y Tecnológica (PICT 20479) for the grants obtained.


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a-Ti and Zr: Diffusion in Materials and its Application: Recent Development”, Materials Transactions JIM, 44 (2003) 2-13.
8. M. J. Iribarren, M. M. Iglesias and F. Dyment, “Diffusion along Grain and Interphase Boundaries in Alpha Zr and Zr-2.5 Wt Pct Nb Alloy”, Met. and Mat. Trans. A, 33 (2002) 797-800.

Contact Details

*C. Corvalán Moya and M. J. Iribarren

Comisión Nacional de Energía Atómica (CNEA)
Av. del Libertador 8250 (1429) Buenos Aires
República Argentina

E-mail: [email protected]r

C. Corvalán Moya and F. Dyment

Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET)

This paper was also published in “Advances in Technology of Materials and Materials Processing Journal, 9[2] (2007) 161-166”.

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